Gaussian measure on a Banach space
When a Gaussian process has continuous (or otherwise nice) sample paths, its law is naturally a probability measure on a function space — an infinite-dimensional Banach space such as C[0, 1]. Studying that measure directly, rather than the process coordinate by coordinate, is the geometric/measure-theoretic way to do Gaussian analysis. Wiener measure (the law of Brownian motion on C[0, 1]) is the prototype.
Let E be a separable Banach space with dual E^*. A Borel probability measure mu on E is Gaussian if, for every continuous linear functional L in E^*, the pushforward L_# mu is a one-dimensional Gaussian law on R (possibly degenerate). Equivalently, every L in E^*, viewed as a random variable on (E, mu), is a real Gaussian. The mean is the vector m in E with L(m) = integral L dmu for all L, and the covariance is the bilinear form (L_1, L_2) -> integral (L_1 - L_1(m))(L_2 - L_2(m)) dmu on E^* x E^*. A centered (m = 0) Gaussian measure is determined by this covariance operator. Two facts are central and surprising in infinite dimensions: there is no Lebesgue measure / no translation-invariant reference measure, so Gaussian measures are studied via their own structure; and by Fernique's theorem the norm has a Gaussian tail, integral exp(alpha ||x||^2) dmu(x) < infinity for some alpha > 0, so all moments of ||x|| are finite even though the space is infinite-dimensional.
Why it matters: this is the setting for the Cameron-Martin theorem (which shifts are admissible), the Borell-TIS concentration inequality (the norm/supremum concentrates around its mean), and abstract Wiener space (the canonical triple Cameron-Martin space, Banach space, Gaussian measure). A crucial caveat about dimension: in infinite dimensions two centered Gaussian measures are either equivalent (mutually absolutely continuous) or mutually singular — the Feldman-Hajek dichotomy — there is no middle ground, and most shifts and scalings that look harmless in R^n produce a singular measure. The Cameron-Martin space is exactly the (measure-zero) set of shifts that keep you in the equivalent class.
Wiener measure is the Gaussian measure on E = C[0, 1] (continuous functions vanishing at 0) that is the law of Brownian motion. The coordinate functionals L_t(w) = w(t) are Gaussian with Cov(L_s, L_t) = min(s, t). Fernique guarantees E[exp(alpha (max_t |w(t)|)^2)] < infinity for small alpha, so the maximum of a Brownian path has all moments and a Gaussian-type tail.
Wiener measure on C[0,1] is the canonical Gaussian measure; Fernique's theorem gives its norm a Gaussian tail.
In infinite dimensions two centered Gaussian measures are either equivalent or mutually singular (Feldman-Hajek) — there is no Lebesgue measure and no smooth in-between.